Chapter 13 Overview

• What happens when there is more than one seller, but still there are few enough to affect the market?
• How do their choices affect each other?
• Will they cooperate? Will they compete?
• What happens if their products are imperfect substitutes?

Objectives

• Describing and Measuring Market Structure
• Oligopoly with Homogeneous Products
• Dominant Firm Markets
• Oligopoly with Horizontally Differentiated Products
• Monopolistic Competition
• The Cournot Equilibrium and the Inverse Elasticity Pricing Rule

Measures of Market Structure

• Four-Firm Concentration Ratio (4CR): The share of industry sales revenue accounted for by the four firms with the largest sales revenue in the industry

• Herfindahl–Hirschman Index (HHI): The sum of the squares of the market share of each firm in the industry (\(0 \le HHI \le 10,000\))

• Market Structure	4CR	HHI
  Perfect Competition	Low	Low
  Olygopoly	Intermediate	Intermediate
  Monopoly	100	10,000

Oligopoly with Homogeneous Goods

• A central feature of oligopoly markets: competitive interdependence
  • The decisions of every firm significantly affect the profits of competitors
  • Perfect Competition: No impact of one firm on its rivals
  • Monopolist: No rivals
• Central question of oligopoly theory: how does the close interdependence among firms in the market affect their behavior?

The Cournot Model

• 2 firms (duopoly); one homogeneous good
• Both with identical marginal costs
• Firms choose output (how much to produce)
  • simultaneously
  • non-cooperatively (no collusion)
  • no knowledge of each other’s plan

The Cournot Model

• Inverse demand is downward-slopping and a function of the combined output of the two firms, \(P(Q_1+Q_2)\)
  • Price is not known until both firms have made their output choice
• Each firm will produce the output choice that maximizes its profit based on its expectation of the other firm’s output choice

Residual Demand

• Suppose firm 1 expects that firm 2 will produce \(Q^e_2\)
• If firm 1 produces \(Q_1\), then total output will be \(Q=Q_1+Q_2^e\), and
• and market price will be \(P(Q)=P(Q_1+Q_2^e)\)
• the inverse market demand resulting from holding their rivals’ output constant is the residual demand
  • Ex. Linear Demand \[ P(Q)=a-b(Q_1+Q_2^e)=(a-bQ_2^e)-bQ_1 \]

Residual Demand

Example

• Inverse demand curve \(P(Q)=100-Q\)
• \(MC=10\)
• Firm 1 is Samsung and Firm 2 is SK
• If \(Q=80\), then \(P(80)=20\)
• If \(Q^e_2=50\), then \(P(Q_1+50)=(100-50)-Q_1\)
  • for \(Q_1=30\), \(P(30+50)=20\)

Residual Demand

Best Response

• When choosing output, each firm will act as a monopolist relative to its residual demand

• Firm 1 maximizes its profit by choosing \(Q_1\) that maximizes: \[ \pi_1=P(Q_1+Q_2^e)⋅Q_1−C(Q_1) \]

• Best Response: For any given belief about the output of firm 2, \(Q^e_2\), there is an optimal choice of output for firm 1, \[ Q_1=BR_1(Q^e_2) \]

Reaction Function

• The best response function for Firm 1, \(BR_1(Q_2^e)\), is derived by setting: \[ \frac{\partial \pi_1}{\partial Q_1} = 0 \]

• Example with Linear Demand: Assuming \(P(Q) = a - b(Q_1 + Q_2)\) and constant marginal cost \(MC = c\), the reaction functions can be simplified as: \[ \begin{aligned} Q_1= \frac{a - c}{2b} - \frac{Q_2^e}{2} \;;\; Q_2= \frac{a - c}{2b} - \frac{Q_1^e}{2} \end{aligned} \]

Reaction Function

Example (continued)

• With \(P(Q)=100-Q\) and \(MC=10\)
• \[ Q_1=45-\frac{Q^e_2}{2} \;;\;Q_2=45-\frac{Q^e_1}{2} \]

Rection Function

Cournot Equilibrium

• Cournot Equilibrium: each firm maximizes its profits, given its beliefs about the other firm’s output choice, and those beliefs are confirmed in equilibrium
  • Each firm optimally produces the output its rival expects it to produce
  • No firm has incentives to deviate
• Mathematically, a combination of output choices \((Q_1^*,Q_2^*)\) that satisfy \[ \begin{aligned} Q^*_1= \frac{a - c}{2b} - \frac{Q_2^*}{2} \;;\; Q^*_2= \frac{a - c}{2b} - \frac{Q_1^*}{2} \end{aligned} \]

Cournot Equilibrium

• Cournot Equilibrium for Two Firms: Occurs where the reaction curves of Firm 1 and Firm 2 intersect.

• Equilibrium Output: Solving the system of equations given by the reaction functions yields the equilibrium outputs for both firms: \[ Q_1^* = Q_2^* = \frac{a - c}{3b} \] and the total market output is: \[ Q^* = Q_1^* + Q_2^* = \frac{2(a - c)}{3b} \]

Example (continued)

• Continuing with our example \(P=100-Q\) and \(MC=10\)

• Firm output is \(Q_i=30\), total market output is \(Q=60\)

• So, market price is \(P=40\)

Achieving Equilibrium

In the Cournot equilibrium both firms fully understand their interdependence and have confidence in each other’s rationality

Cournot for N-firms

• For \(N\) identical firms, total output is \(Q=\sum_i^N Q_i\)

• Firm \(i\) solves \[ \max_{Q_i} P(Q)⋅Q_i-C(Q_i) \qquad(1)\]

• For linear demand, and \(MC=c\), each firm \(i\) will produce \[ Q^*_i = \frac{1}{N+1}\frac{a - c}{b} \]

Cournot for N-firms

• Total market output and market price will be \[ Q^*=\sum_i Q^*_i = \frac{N}{N+1}\frac{(a - c)}{b} \] \[ P^*=a-b\left(\frac{N}{N+1}\frac{(a - c)}{b} \right) = \frac{a+Nc}{(N+1)} \]

Comparing Market Structures

Comparing Market Structures

• Monopoly: A single firm controls the entire market, producing half the competitive output and selling at a higher price.
• Cournot Duopoly: Two firms share the market, leading to increased output and lower prices compared to monopoly.
• Perfect Competition: An infinite number of firms produce at marginal cost, leading to the highest output and lowest price.

Comparing Market Structures

Cournot vs Monopoly

By independently maximizing their own profits, firms produce more total output than they would if they collusively maximized industry profits (Monopoly).

Cournot and Elasticity of Demand

• FOC of Equation 1 gives \[ P(Q)+\frac{\partial P}{\partial Q}⋅Q_i=MC(Q_i) \]

• It can be shown that \[ P(Q)\left[1-\frac{1}{|\epsilon_{Q,P}|/s_i}\right]=MC(Q_i) \] where \(s_i=Q_i/Q\) is the market share

Cournot and Elasticity of Demand

• \(|\epsilon_{Q,P}|/s_i\), the elasticity of the demand curve facing the firm:
  • the smaller the market share of the firm, the more elastic the demand curve it faces

• \(s_i\)	Demand Curve	Condition	Market Structure
  0	Flat	\(P=MC\)	Perfect Competition
  1	Market Demand	\(P(1-\frac{1}{|\epsilon_{Q,P}|})=MC\)	Monopoly